在MATLAB中编写微分方程 dy/dx＝xy, 当 x＝0 时 y＝1+1/5, x 属于 0～3 之间，编写积分程序，包括欧 拉数值积分程序，预报校正数字积分程序、4 阶龙格库塔积分程序，它们的积分步长分别取0.01，0.1, 0.5， 绘制积分结果曲线

好的，我知道了。下面是代码实现：

% 定义微分方程
f = @(x, y) x*y;

% 初始条件
x0 = 0;
y0 = 1.2;

% 定义积分步长
h1 = 0.01;
h2 = 0.1;
h3 = 0.5;

% 定义积分区间
xspan = [0, 3];

% 使用ode45求解微分方程
[x, y] = ode45(f, xspan, y0);

% 绘制微分方程解的图像
plot(x, y, 'r-', 'LineWidth', 2);
xlabel('x')
ylabel('y')
title('Solution of dy/dx = xy')

% 欧拉数值积分程序
% 积分步长为0.01
n1 = (xspan(2) - xspan(1)) / h1;
xe1 = linspace(xspan(1), xspan(2), n1+1);
ye1 = zeros(size(xe1));
ye1(1) = y0;
for i = 1:n1
    ye1(i+1) = ye1(i) + h1 * f(xe1(i), ye1(i));
end

% 预报校正数字积分程序
% 积分步长为0.1
n2 = (xspan(2) - xspan(1)) / h2;
xe2 = linspace(xspan(1), xspan(2), n2+1);
ye2 = zeros(size(xe2));
ye2(1) = y0;
for i = 1:n2
    k1 = f(xe2(i), ye2(i));
    k2 = f(xe2(i)+h2/2, ye2(i)+h2/2*k1);
    k3 = f(xe2(i)+h2/2, ye2(i)+h2/2*k2);
    k4 = f(xe2(i)+h2, ye2(i)+h2*k3);
    ye2(i+1) = ye2(i) + h2/6*(k1+2*k2+2*k3+k4);
end

% 4阶龙格库塔积分程序
% 积分步长为0.5
n3 = (xspan(2) - xspan(1)) / h3;
xe3 = linspace(xspan(1), xspan(2), n3+1);
ye3 = zeros(size(xe3));
ye3(1) = y0;
for i = 1:n3
    k1 = f(xe3(i), ye3(i));
    k2 = f(xe3(i)+h3/2, ye3(i)+h3/2*k1);
    k3 = f(xe3(i)+h3/2, ye3(i)+h3/2*k2);
    k4 = f(xe3(i)+h3, ye3(i)+h3*k3);
    ye3(i+1) = ye3(i) + h3/6*(k1+2*k2+2*k3+k4);
end

% 绘制积分结果曲线
hold on
plot(xe1, ye1, 'g--', 'LineWidth', 2);
plot(xe2, ye2, 'b:', 'LineWidth', 2);
plot(xe3, ye3, 'm-.', 'LineWidth', 2);
legend('ode45', 'Euler', 'Predictor-Corrector', 'RK4');

运行程序后，会得到以下图像: